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Another Look at Differential-Linear Attacks

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Selected Areas in Cryptography (SAC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13742))

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Abstract

Differential-Linear (DL) cryptanalysis is a well known cryptanalytic technique that combines differential and linear cryptanalysis. Over the years, multiple techniques were proposed to increase its strength. Two recent ones are: The partitioning technique by Leurent and the use of neutral bits adapted by Beierle et al. to DL cryptanalysis.

In this paper we compare these techniques and discuss the possibility of using them together to achieve the best possible DL attacks. We study the combination of these two techniques and show that in many cases they are indeed compatible. We demonstrate the strength of the combination in two ways. First, we present the first DL attack on 4-round Xoodyak and an extension to 5-round in the related key model. We show that the attacks are possible only by using these two techniques simultaneously. In addition, using the combination of the two techniques we improve a DL attack on 9-round DES. We show that the partitioning technique mainly reduces the time complexity, and the use of neutral bits mainly reduces the data complexity, while the combination of them reduces both the time and data complexities.

The first author was supported in part by the Center for Cyber, Law, and Policy in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office and by the Israeli Science Foundation through grants No. 880/18 and 3380/19.

The second author was supported by the European Research Council under the ERC starting grant agreement n. 757731 (LightCrypt), by the BIU Center for Research in Applied Cryptography and Cyber Security in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office, and by the President Scholarship for Ph.D. students at the Bar-Ilan University.

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Notes

  1. 1.

    We note that some works divide the cipher into three sub-ciphers \(E=E_1\circ E_m\circ E_0\) (e.g., [3, 11,12,13, 25, 26]). This is mostly done to better understand the transition between the two main sub-ciphers \(E_0,E_1\) and most importantly the dependencies between the two sub-ciphers. The emphasis of this paper is the external rounds (rather the internal rounds and the transition). Our results are independent of these works and thus we use the simpler description of DL attacks. We note that both partition and neutral bits may still result in subtle dependencies which may impact the transition, and hence we experimentally verified our results whenever possible.

  2. 2.

    A similar idea is used in the chosen-plaintext linear attack of Knudsen and Mathiassen on DES [20].

  3. 3.

    More precisely, we show that each subset in the partitioning (which is defined according to the key material) determines a good value for the non-neutral bits. Without the combination, the distinguisher cannot be used for a key recovery attack.

  4. 4.

    The partitioning can be applied to plaintexts, ciphertexts, or any other criteria. For example, in [22] the partitioning is performed also according to the values of the ciphertexts.

  5. 5.

    It should be noted that not always all the vectors in the linear subspace are neutral (see [7] that discusses such examples). However, in all of the cases discussed here this is the scenario.

  6. 6.

    Similar issue also affected chosen-plaintext linear cryptanalysis [20].

  7. 7.

    The probability should be significantly higher than the differential’s probability.

  8. 8.

    This idea was used in [20].

  9. 9.

    Liu et al. [24] present a 4-round rotational DL distinguisher, with the highest possible bias of \(\frac{1}{2}\), without any attack that uses it. We give in the ePrint version the rotational DL distinguisher used by Liu et al. and recall that rotational DL distinguisher is not a DL distinguisher.

  10. 10.

    In detail, for each \(0\le i<128\), when the input difference is \((0,e_i,e_i)\), the best results occurs for the output mask \((0,e_{32\cdot \lfloor \frac{i}{32}\rfloor + (15+i\pmod {32})},0)\). It should be noted that since the mask is in the second plane and only the first 64 bits of this plane are visible, we can not use all the 128 characteristics, but only the 64 characteristics for which \(0\le i<64\). However, this fact does not impact our analysis.

  11. 11.

    All the experiments can be found in https://github.com/ArielWeizman/AW/blob/master/Xoodoo.

  12. 12.

    In detail, for each \(0\le i < 128\), when the input difference is \((e_i,0,e_i)\), the best results occur for the output mask \((e_i,0,0)\).

  13. 13.

    We note that the time complexity is about \(2^{12.82}\cdot 2\cdot 2^6\) one S-box evaluations, which are equivalent to about \(2^{13.82}\) 8-round encryptions.

  14. 14.

    All the experiments can be found in https://github.com/ArielWeizman/AW/blob/master/DES..

  15. 15.

    For attacks that needed more plaintext pairs, we refer the reader to the chosen plaintext linear cryptanalysis techniques suggested Knudsen and Mathiassen [20].

  16. 16.

    Since each plaintext pair passes the differential characteristic has zero difference in the bits masked by \(\lambda _M\) (i.e., \((P\oplus P')\cdot \lambda _M = \varOmega _M \cdot \lambda _M = 0\)), the sign of the bias is necessarily positive.

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Authors and Affiliations

  1. Computer Science Department, University of Haifa, Haifa, Israel

    Orr Dunkelman

  2. Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel

    Ariel Weizman

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  1. Inria and École Polytechnique, Institut Polytechnique de Paris, Palaiseau, France

    Benjamin Smith

  2. Electrical and Computer Engineering, University of Windsor, Windsor, ON, Canada

    Huapeng Wu

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Dunkelman, O., Weizman, A. (2024). Another Look at Differential-Linear Attacks. In: Smith, B., Wu, H. (eds) Selected Areas in Cryptography. SAC 2022. Lecture Notes in Computer Science, vol 13742. Springer, Cham. https://doi.org/10.1007/978-3-031-58411-4_6

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